The famous paintings by Dutch painter Piet Mondrian, featuring iconic coloured rectangles, are not only one of the greatest expressions and representations of modern abstract art that has inspired generations of aspiring artists, they are also the basis of an interesting problem or mathematical puzzle.
Mondrian’s Mathematical Problem consists of dividing a grid of dimensions n x n, into rectangles and squares of whole and incongruent sides (i.e. no two are equal), so that the difference between the surface of the largest and smallest rectangle is as small as possible. That subtraction will give the score.
This statement, formulated in this way in the abstract, may be difficult to understand, but it is very easily seen with a concrete case. For example, take the 4×4 grid. One possible solution is to divide the grid into two rectangles of 3×4 and 1×4, which gives a score of 8. One way to improve the result is to divide the grid into 3 rectangles, which lowers the score to 6.
There is still a better solution for a 4×4 square, an optimal distribution that gives a score of 4. What is it?
It was in 1915, after having absorbed and assimilated the principles of art movements such as impressionism, expressionism and cubism, that Piet Mondrian began to produce his famous paintings. These works sublimated the abstraction and simplification of shapes to limit the elements to straight lines and rectangles and the colours used to primary (red, yellow and blue) and achromatic (grey, white and black). With this pictorial vocabulary, the painter tried to reflect the balance of opposites —lines vs. surfaces; horizontal vs. vertical forms; vivid colours vs. the absence of colour— that governed nature and the entire universe and constituted its essence and spirit.
This geometric distribution of Mondrian’s paintings —so studied and based on shapes that were opposing and at the same time complementary— captured the attention of mathematicians who would pose this challenge, which becomes more complicated as the grid becomes larger:
Here is the best solution for the case of a 5×5 square, which allows a score of 4.
In the case of a 6×6 square, the minimum possible score is 5. What is the solution that allows this value to be reached?
And what is the optimal solution for an 8×8 square?
Looking for a mathematical shortcut
Every since the Mondrian Problem was raised, mathematicians have tried and continue to look for a general method or algorithm that would give the solution for the infinite possible cases. While this has not yet been achieved, it would be a very useful practical tool in fields such as the distribution and optimisation of spaces or of packaging.
The main difficulty involved is that, by increasing the dimensions of the square, no pattern or structure has been identified in the solutions (something that could serve as a clue or starting point towards obtaining the overall solution). In fact, the contrary is true; although as the value of n increases, the optimal score does seem to stabilize rather than grow, it continues to fluctuate in both directions. Larger squares may have scores lower than those preceding it, but also higher, or the same. And there is even more variability in the number of rectangles needed to get the optimal score in each case.
In the absence of finding this mathematical shortcut, until now the solutions obtained for grids of size n x n are reached by means of “brute force” approximations, that is to say, through algorithms that test in each case —for each value of n— the greatest possible number of solutions and calculate the score of each one of them.
Below is the best solution for a 10×10 square, whose score is 8.
On the other hand, the 11×11 and 12×12 squares have optimal solutions that allow scores even lower than 8. What are the solutions for each case? What is the minimum score that can be achieved in each case?
In its most playful aspect, the Mondrian Problem is on the face of it a simple puzzle that is inexhaustible, since it can be enlarged and complicated to infinity, as long as the dimensions of the grid are increased. That (same) lack of structure or regularity that exasperates mathematicians is what makes it especially attractive and addictive.
And if all of the above were not enough, aficionados can also choose to go one step further and introduce an “artistic” element or condition, such as colouring the resulting distribution using the least number of colours possible and requiring that rectangles that share vertexes and/or edges cannot have the same colour. This allows us to obtain solutions as eye-catching as those exhibited as examples… and to get a little closer to Mondrian’s work.